Question 114
$\{a_n\}_{n \ge 0}$ and $\{b_n\}_{n \ge 0}$ are two sequences of integers with $a_i, b_i \in \{0, 1, 2, \dots, 9\}$. There is an integer $M$ such that $a_n, b_n \ne 0$ for all $n \ge M$, and for each $n \ge 0$
$$
(\overline{a_n \cdots a_1 a_0})^2 + 999 \mid (\overline{b_n \cdots b_1 b_0})^2 + 999
$$
(Note that $(\overline{x_n x_{n-1} \dots x_0}) = 10^n \times x_n + \dots + 10 \times x_1 + x_0$.)
Prove that $a_n = b_n$ for $n \ge 0$.
$$
(\overline{a_n \cdots a_1 a_0})^2 + 999 \mid (\overline{b_n \cdots b_1 b_0})^2 + 999
$$
(Note that $(\overline{x_n x_{n-1} \dots x_0}) = 10^n \times x_n + \dots + 10 \times x_1 + x_0$.)
Prove that $a_n = b_n$ for $n \ge 0$.